Recurrent tensors and holonomy group

Abstract

Let M be a connected C∞. A method is being introduced here to study the action of the holonomy group and the restricted holonomy group of Γ on a recurrent tensor. The main result of this paper is that if the recurrence covector W of a recurrent tensor S on M is an exact form then the tensor S is invariant under the holonomy group of Γ and if W is a closed form then S is invariant under the restricted holonomy group of Γ. In the last section, this result is applied to some particular cases including the case of a riemannian manifold with recurrent curvature.